Readings Reflections
The readings reflections have two main purposes:
1) to hold you accountable for careful reading of and reflection on the readings assigned in class; and
2) to provide you with a record of what you've learned and thought about as a result of the readings.
The readings reflections will be evaluated using the following criteria:
completeness and timeliness of the entries;
comprehension of the main ideas of the readings; and
depth and quality of integration of the ideas with your own thinking.
Submit your readings reflection before reading anyone's on the Wiki page and then paste it into the existing reflection page for that current reading. This is due by Wed 9/28 by 11:30pm.Thanks Kaitlin for reminding me and sorry this is late.
No discussion is required but feel free to post if you want to. This is not for extra credit, just if you want to note something.
In the first paragraph of this article the authors made a close connection between reflection on teaching done by teachers and their professional growth (1-1). The people at Quesar suggest that math teachers focus on tasks given in class to reflect on to improve their teaching. They defined a task to be a segment of classroom activity that is devoted to the development of a particular mathematical idea (2-1).
The tasks framework starts with the task as a textbook presents it. Then it moves to tasks as set up by the teacher, and finally the framework deals wit tasks as implemented by students. The article mentions a few instances where teachers’ lessons didn’t go according to the plan. At one point it says that “students appeared overwhelmed” by the number of choices need to be made to begin an assignment (4-3). When students encounter challenges they are many times indecisive in choosing an approach to attack the problem. We need to give students confidence without directly guiding them to the answer. I also think that students need to be more independent in problem solving, but a teacher will need to design lessons that will improve their independence yet remain rewarding if their attempt fails.
Another case in the article talked about a student asking “how do you do this” (5-1) The teacher let it go for awhile and then gave in and gave direction to the class on how to do the problem. This was later repeated but the teacher refrained from giving them advice on what to do. The framework helped him realize his mistake in the first lesson. When he repeated the lesson he was able to improve his teaching. The framework is only a tool of reflection. The article claims that reflection leads to professional growth, which will lead to better teaching. I can agree that reflection is good to improve teaching, but I thought their framework was too rigid. I don’t think that reflection can be put to some formula. I think that it would be difficult to implement this framework into reflection practices in the classroom. I think having peers (specifically teachers with experience) in the same field observe and suggest ideas to you would be a tremendous help to young teachers. I don’t know how feasible this is because schools are giving teachers more classes to teach, which means less time to plan lessons during the school day. I would personally look for other suggestions on reflection before implementing this framework into my classroom. Mike F
The article, “Mathematical Tasks as a Framework for Reflection: From Research to Practice,” aims to discuss how teachers can reflect on and evaluate their classroom experiences (268-1). I think it is important for teachers to take a step back and reflect on things that are occurring in their classrooms. If we take a good hard look at what is going on we might discover things such as more struggling students than we thought and so on. As promising as they sounds, unfortunately it is not always easy to figure out what we as teachers need to focus on when reflecting and evaluating (268-2). There is good news however, the QUASAR Project has discovered that focusing on mathematical tasks and their phases of classroom use can help teachers during the process of reflecting (268-2).
A task, defined in the article as a period of class activity that is focuses on the progression of a particular mathematical idea (268-2). To me a task seems to be a problem that would be given to a group of students to solve without giving them any specific way of solving the problem. In this respect students can really dive into the problem and discover on their own a way to solve it. This again leads to the idea of cooperative learning which I think is a good way to learning for students.
The article goes on to describe task phases (270-3) in which implementation is the most important stage because it involves the students thinking for themselves and being creative with how to solve problems instead of simply plugging numbers into a formula.
I particularly enjoy the task given to the students in, “The Case of Ms. Bradford” (271-1). It seemed like something the students would really enjoy and get them actively involved but also something they could learn from. However, I also saw that how it could be overwhelming for the students. I think with the right changes it could have been what it was intended. For example, do not give the students as much freedom, I understand it was so the students could come up with ideas on their own but I also believe they need a starting place and apparently so by how overwhelmed they were. I think something of that magnitude needs to be worked up too. Like the article stated, many students are not used to being given that amount of freedom and it slowly needs to be worked up too (271-4).
The discussion of the figure 1 (273-2) I felt that Ron gave his students too much information instead of letting them figure it out. For example he told them to find what percent was 6 shaded squares on a 10X4 rectangle. Instead I think maybe give them a percent and have them figure out the number of squares that would equal. I feel that in working this way it was would been more of a challenge instead of finding the “tried and true” method.
Overall, I think it is important for teachers to reflect on their lessons and find out what is working and what is not. Also once teachers are more comfortable in their classrooms perhaps video tape lessons and discuss them with future teachers to get some advice. I think its hard sometimes to see what is going wrong with a lesson plan that you worked really hard on, which is where another pair of eyes might help.
Kaitlin Froehlke
This article presents teachers with a framework they can use as they reflect on classroom experiences, making them aware of how they teach and how students are learning (1-1). The main focus for the framework is on mathematical tasks (where a task is defined as “a segment of classroom activity that is devoted to the development of a particular mathematical idea” (1-2)).
The day-after-day experience of tasks provides the foundation for what students believe about the nature of mathematics (2-1). Additionally, I believe that as future teachers we have been conditioned by tasks that we did in middle and high school; our own ideas about mathematics come from this cumulative experience. Therefore, we must understand the drastic effects the collective experiences of students in the classroom have in shaping their ideas and perceptions and utilize tasks in our classrooms that cause students to see mathematics for its logic, beauty, and complexity.
There are different levels of demands in certain tasks; lower-level demands include memorization and procedures without connections, whereas procedures with connections and actually doing mathematics fall into the higher-level (3-1). Higher-level demands allow students to attach meaning to their work and apply their knowledge with an answer that makes sense and can be justified (3-2). Clearly, we as teachers should aim to set up tasks that employ the higher level, and this fulfills the goals of NCTM’s standards.
First, as we have seen, teachers should focus on tasks. Next, they should concentrate on task phases, which include the way the tasks appear in instructional materials, the way teachers set them up, and the way students go about working on the task (3-3). These are vital factors in influencing what students actually learn. Sometimes, as the author points out through the case studies in which the framework was applied, tasks (even ones that are set up at higher levels) evolve and do not produce the intended results (3-4). The way students actually carry out the task can change the level of thinking dramatically (3-4). In order to remedy this, teachers apply the framework and use it for reflection.
In keeping with the past several articles we have read, I noticed a few consistent themes: the power of teachers’ questioning and listening skills in shaping students’ learning, our tendency to give answers and help too quickly, and building on prior knowledge (which is reminiscent of part of the learning standard from NCTM). As with the other articles we read recently, this article stressed how this approach is not easy. For the teachers in the case studies, it was easy to show students how to set up the problem and consequently bring a task to a lower level than intended. According to students’ prior knowledge, when they do not understand something, they wait for the teacher to show them how to do it (4-4). “[B]y succumbing to the students’ requests of ‘how to do it’”, we reduce or eliminate “the challenging, sense-making aspects of the ask, thereby robbing students of the opportunity to develop thinking and reasoning skills and meaningful mathematical understandings.” (5-2); I certainly agree.
As we consider mathematical tasks as a way to reflect on our teaching, our reflection becomes more focused and can lead to significant, beneficial change. Using this reflection framework with other teachers yields even better results through cooperation, and their insights can be very helpful as well. As I begin my teaching career and start, even now, to reflect on my teaching, an integral part of this will be to evaluate the mathematical tasks and how the tasks are carried out in my classroom. Without considering the tasks, I may have no idea where to begin to improve my teaching; my students’ learning would suffer from this. Therefore, in order to help my students understand mathematics as well as they possibly can, I must evaluate and reflect upon mathematical tasks in my classroom.
Mandi Mills
Stein and Smith’s article “Mathematical Tasks as a Framework for Reflection,” encourages teachers to reflect upon their own teaching styles in order to improve their abilities as an educator. They point out, however, that simply asking teachers to reflect is too vague, and finding a focal point upon which to base our reflections is the most difficult part of the process (1-2).
This is why Smith and Stein suggest that our focus be on mathematical tasks in the classroom. Students expand their mathematical abilities while working with numerical problem solving tasks (2-2), rather than listening to a lecture or following a set procedure. Because most of their growth as mathematicians happens during their work on mathematical tasks, our ability to design and carry out appropriately challenging tasks is essential. If we reflect upon our teaching abilities in regards to creating these challenging tasks, than we keep our improvement as teachers centered on students’ learning.
The authors go on to suggest that we evaluate our effectiveness in the classroom with QUASAR’s framework best by reviewing previous lessons with colleagues (7-3). I thought this was a clever notion as it mirrors how we group students together in hopes that they will help each other. By working with each other as a community of teachers we will be able to find which of the tasks that we create are already High-Level and which need adjustment to provide students with a better opportunity to think about the problem and reason out an answer that makes sense for them.
In the case of Ms. Bradford, she dealt with a problem that I, myself, have concerns about. How are we to know what tasks are too High-Level? Her tape-roll-toss task lacked direction for the students, who felt overwhelmed by the amount of choices (4-3). This sort of anecdote worries me because while we may plan a lesson that we feel genuinely inspires students to stretch their understanding, if it is not implemented with enough direction than it can fall flat. Though we have discussed at length the benefits of a classroom that minimizes direction, too much open-endedness can leave students feeling frustrated and discouraged by their lack of understanding.
As educators, we must be able to find the balance between too much and too little direction in the classroom. By reflecting on our lessons, with the focus on student learning through mathematical tasks, we can improve the overall atmosphere of our classrooms to one in which students feel capable and interested.
Valerie Gipper
The article “Mathematical Tasks as a Framework for Reflection: From Research to Practice” introduced the idea of tasks used in classrooms. When I was reading this article, I couldn’t help but notice much of what the authors were talking about seemed to be similar to what we have been reading: students need to work things out for themselves. Examples were shown of two different types of tasks. The first was lower-level tasks. These entailed the students memorizing the use of algorithms and computation in addition to doing multiple of these exercises in one task (3-1). I can remember learning math in this type of way. I recall being given plenty of worksheets which said to find the decimal of this or what percent is this fraction? This is the type of tasks we want to avoid with our students. The other type of task, the better choice, is high-level. These allow students to go though procedures like they would in lower-level tasks but also building mathematical connections as they work though it (3-2). In the example they showed in figure 1, the students were asked to use pictorials to explore the concepts of percents, decimals, and fractions. This is considered a high-level task where as, shown also in figure 1; the lower level task is simply computation of fractions and such. The case of Ron Castleman was very interesting and proved to be great insight to the differences between the two tasks. In the first part of this case, Ron attempted to have a lesson similar to that shown in figure 1 in his classroom. The idea was for his students to use a rectangle to find percents, decimals, and fractions. After he started the class he ended up giving his students hints on how to find the answer without using the rectangle. As one of Ron’s colleagues noted, by allowing the students to bypass using the rectangle to find the information, they were completely losing the meaning of a decimal, fraction, and percent (5-3). When I am in a math class and do not understand something my first instinct is to go to the teacher and ask for help. What I gained from this article is the teacher telling a student a huge hint is not helpful. I look back at times when I have asked the teacher for assistance and I realize they either end up doing the entire problem for me or at least give me so many hints I can do it without any more trouble. For the purpose of getting the assignment done this is good for the student but that is where it stops. I don’t remember what the concepts were behind those problems which I asked for. I also have no idea why some of the mathematical techniques work. All I know was I finished the problem. Ron learned to fix this problem by redirecting the students to the original point of the problem, use the rectangle and this helped the students work on their problem solving skills as well as allow them to expand on their connections between mathematical concepts (5-6). Towards the end of the article, teachers observing one another was discussed. This is a great way to get feedback on how things are going in your classroom (8-2). In addition, the teachers who are being observed will be able to learn from their mistakes and know what seemed to work and what didn’t (8-3). The idea of your co-workers observing and taking notes on what you are doing holds you accountable for actually making sure you are supplying your students with a high level task. In addition, the teacher who is observing you should be able to see where your weaknesses are in the process which is a good to know. In addition, they talked about teachers observing themselves. The suggest videotaping yourself, which allows you to reflect on your own time in an objective manner (8-4). Either form of observation seems to give the same results. The teachers are getting feedback and the students will have a better experience because of it. Katey Cook
The article, “Mathematical Tasks as a Framework for Reflection” illustrates the point that it is beneficial for teachers or educators to reflect not only on their students’ progress but on their own as well. (1-1) This may seem to be common sense to most but the idea goes a bit deeper than just common sense. Most teachers will have an informal thought of what went well and what needed more work but rarely do they have an in-depth conversation with a colleague for instance to discuss how to improve their teaching style.
So often as we discuss in class mathematics instruction follows the same traditional path where students memorize steps or procedures but do not gain a full understanding of the main concept. We saw evidence of this in class on Tuesday. Fred knew how to perform the operation algebraically but until that light bulb went off he didn’t quite grasp the whole concept. The author found her students were doing the same thing, memorizing operations rather than thinking about the task as its laid out.(2-1)
In the case of Ms. Bradford, what she thought to be a complex lesson plan that would engage her students while having them think at a higher level of problem solving failed during implementation. (4-2) She found that posing an open ended question to her students was a new and scary experience for them which did not help her execute her plan. Although her lesson plan was a good one she adapted quickly so that the students didn’t get dejected and then later reflected on the lesson with colleagues and found other ways to implement her lesson.(4-3) This shows the resolve a teacher needs to have. Realize when a lesson is failing, adapt that lesson so that the students don’t lose interest, and reflect on the lesson and make changes as needed.
Mr. Castlemen’s story is quite similar although I really enjoyed seeing the progression of the lesson plan. At first he laid out an open ended question for the students but when the students quickly began to ask questions he began to fall into the deep dark pit of straight answering those questions.(5-2) After he had felt his lesson failed he went to colleagues and friends to discuss and diagnose the problem with his lesson. The problem became clear but the solution would be the hard part. He decided that this time he would hold off on answering questions and instead ask questions in return to get his students thinking about the problem in another light. (5-6) Although this task took much longer than he had anticipated the outcome of the majority of his students comprehending the material was well worth it.(5-7)
As we become teachers in our own classroom this lesson is one that should be learned early. Not every lesson will work as planned and not every student will understand the concepts we present immediately. It is integral to the students and the educator’s success that the educator constantly reflects on lessons given to improve lessons that still need to be addressed. I think colleagues in our future schools as well as those met during our classes at WMU will be an invaluable resource to an educator who constantly wants to improve. I hope that’s all our goals. Christopher Cardon
In the article, “Mathematical Tasks”, Stein and Smith discuss the important aspect of reflecting upon lessons and how students are responding. I feel reflection is an essential part of a teacher’s career. If the person’s goal is to be an effective teacher, then being able to critique and learn from your own teaching methods gets you one step closer to that goal. Not only does it benefit the teacher by making her better, but more importantly the students will benefit because they will be experiencing a higher-quality education.
Stein and Smith cover a framework for reflection and the different ways it can be used (2-1). The framework is based on mathematical tasks that are focused on teaching a specific idea. The task can be completed by lower-level or higher-level approaches depending on what the teacher’s goal is for the lesson (3-2). When I was in middle school, I can only remember having low-level demand problems placed in front of me. My teachers back then were more focused on quantity over quality, and it didn’t really matter if we were developing thinking skills in the process (5-3). Now I am conditioned to want to work on lower-level demand problems because I was never introduced to higher-level work. I realize now what an impact these higher-level approaches can have on a student. Not only do they allow students to build connections between mathematical concepts (3-2), but it increases students’ confidence in mathematics. A student who is familiar with higher-level tasks will be more willing to try numerous strategies to obtain a correct answer as opposed to giving up after one try. Whereas, the lower-level task student that is introduced with a higher-level task would probably get frustrated if she couldn’t figure the answer out right away. I am basing that conjecture on articles that we have read and also on my own experience.
Ms. Bradford tried to apply the mathematical tasks framework in her classroom with the tape-roll toss problem (4-2). The students became overwhelmed with the task and what was expected of them, and I actually did too. Just reading about the tape-roll toss made me nervous. If I had been in that classroom, I would have reacted the same way as all those students because of my lack of experience with open-ended tasks. Students that have been trained to solve lower-level tasks become uncomfortable when presented with a problem that they don’t know how to solve (4-4). So I plan on incorporating higher-level tasks into my curriculum. That’s not the way I was taught, but I am realizing there are much more beneficial ways to teach math.
Hailey McDonell
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The article “Mathematical Tasks as a Framework for Reflection: From Research to Practice” stresses the importance of constant self- or colleague-evaluation of teachers. This continual reflection makes us aware of how we are teaching. Is it effective? Is it confusing? They have provided us with a “framework for reflection based on the mathematical tasks used during classroom instruction and the ways in which it has been used by teachers,” (269-1). They have created this because they feel that task form the basis for learning and are varied in level of demand (269-2) The steps in The Mathematical Task Framework include curricular materials, set up by teacher, implemented by students, and when learning occurs (270-4).
The authors go on to give cases to reflect upon based on their research for the QUASAR Project. First, Ms. Bradford challenged her class to create a game board for an upcoming carnival at their school for a ring toss-style game (271-2). She supplied them with various different materials to use and set them free to create. Unfortunately, most of her class was not accustomed to this time of mathematical exploration and appeared overwhelmed by too many options. This example sounds so familiar to many creative projects I had encountered back in high school. For example, in English everyone had to write a research paper, but the topic of the paper could be practically anything school appropriate. Most kids spent the first week complaining that the instructions were too vague and nobody had a clue where to start or what to research. Unlike Ms. Bradford’s choice to guide students in their design (271-3) my teacher stayed neutral and forced us to create something all on our own.
The article then gives the case or Mr. Castleman who felt like his class answered one of his tasks too quickly (271-7). They decided to apply the Mathematics Task Framework to think about the lesson. After observing a videotape of the class, they saw that Mr. Castleman started to guide his class just as Ms. Bradford had. Later, he used the same lesson with a different class and tried to apply some tactics learned from the Framework. This time “students showed no inclination to even check the plausibility of the answers that they came up with against the diagram,” (272-3). This Mathematical Task Framework allows us to use more complex and meaningful tasks that are better for students to make sense of mathematics.
I think videotaping your own teaching to reflect upon what your students are thinking while working on a task is a worthwhile self-assessment tool, and if possible ask a trusted colleague to give you constructive feedback in real time. Real-time observation can benefit you both because it forces your colleague to give critical feedback and not gloss over the important areas that need growth (275-1). Another important tid-bit I got out of this article is the suggestion to ask students to explain their varying paths to the same answer to articulate that there isn’t always ONE right way to answer a mathematical task (272-7).
Tori Ward
The article “Mathematical Tasks” is about the importance of teachers reflecting their work to better their performance in the classroom. According to the professional standards for teaching mathematics, a primary factor in teachers’ professional growth is the extent to which they “reflect on learning and teaching individually and with colleagues”. 268-1 Self reflection is important factor because it helps you as individual acknowledge the things you are not doing so well to help student out in the classroom. Reflection with colleagues is important because they can give helpful strategies for you to use in the classroom. Teachers’ reflecting on their classroom experiences is a way to make teachers aware of how they teach and how their students are thriving within the learning environment that has been provided. Reflection holds the key to improving ones teaching as well as to sustaining lifelong professional development. 268-1 Although self reflection is a good thing a question to consider is figuring out what to focus on that needs improvement.
Therefore in this article there is a framework provided for reflection based on the mathematical tasks used during classroom instruction. A task can involve several related problems that can take up an entire class period. 269-1 The focus on mathematical tasks is built from the idea that the task used in the classroom form the basis for student learning.269-2 All tasks lead to student thinking. The day in and day out cumulative effect of classroom based tasks leads to the development of students’ implicit ideas about the nature of mathematics about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so. 269-2 There are three phases through which a tasks goes through: first, as they appear in curricular or instructional materials on the printed pages of textbooks, and so on; next as they are set up or announced by the teacher; and finally as they are actually implemented by students in the classroom. 270-5 All these, are especially viewed as important influences on what students actually learn. After teachers learned about the framework, they began to use it as a lens for reflecting on their own instruction and as a shared language for discussing instruction with their colleagues. A teacher named Theresa said that the framework gave her a language for describing events that has occurred in her classroom and for understanding why things may not have worked out as she envisioned that they would when she did a reflection on her classroom. 271-5
When using the framework it should draw attention to what students are actually doing and thinking about during mathematics lessons. The focus on student thinking helps the teacher adjust instruction to be more responsive to, and supportive of, students attempts to reason and make sense of mathematics. 274-2 Factors associated with the maintenance of high level cognitive demands are scaffolding students thinking, that they are monitoring their own progress, sufficient time is allowed for explorations and tasks build on students prior knowledge. More tasks are also listed in the article but these are just a few to name. 274-5 Factors associated with the decline of high level cognitive demands are classroom management problems or the teacher shifts the emphasis from meaning, or understanding to the correctness or completeness of the answer. More tasks are also listed in the article however these are just a few to name. 274-5 Students who performed the best on project based measures of reasoning and problem solving were in classrooms in which tasks were more likely to be set up and implemented at high levels of cognitive demand. These students performed substantial learning gains which is exactly what the NCTM’s professional teaching standards advocated. As a secondary educator I must know how to measure sit and focus on my teaching methods in order to provide a way for my students to obtain a higher level of thinking. With this framework I am able to ask myself what I may have to focus on in order to reach this goal.
Fredrick Martin
Mathematical Tasks as a Framework for Reflection is an article that provided examples of fidelity to the curriculum and your personal planned task. I felt that it is trying to allow us to realize that we must be intentional in planning and thinking about how to help our students without doing it for them in basic terms (6-3). It makes for an extremely helpful rubric when we are trying to evaluate ourselves for the sake of being efficient and maintaining the ability to change to meet the needs of your students. I also realized that there is a certain amount of cooperation amongst the staff that was displayed in the examples of the two teachers whose stories were shared. The three phases (3-4) should probably be examined for consistency in each phase a task goes through. I usually find that if I am inconsistent in my planning or in my delivery it seems that everything I planned on doing goes out of the window and the classroom is free to really do as they please.
This article pushes the need to be intentional and striking the right balance when it comes to doing things in the classroom. Being intentional requires you to think of what kind of cognitive tasks you are requiring of your students and what questions you will ask them (5-5). You don’t want to give them the answers but at the same time you want them to work on the task without giving up on it too easily. Which brings us to how the right balance of different components comes into play. As a teacher we must constantly re-evaluate how much time we need on a task, what kind, if any, extra measures are needed and on and on all day long. I have heard that a teacher makes a decision every 20 seconds. If this is true that doesn’t really leave any time to evaluate themselves. In this case it is nice to have another teacher to be able to critique you on your performance. It is precisely this framework that allows teachers to get the best of both worlds, not only improving their classrooms but also improving themselves.
Denise Slate
This article follows our class’s current theme of how we can incorporate higher-level thinking into our lesson plans. It affirms that direct teaching or lecturing and routine lower level tasks, like memorizing algorithms, are a teaching method of the past. Student centered learning is the focus for teachers today, and the mathematical task framework appears to be a solid method to evaluate your progress transforming your style to a more ideal method.
The framework is basically an organized way to reflect on your classroom experiences in a way that helps you, as a teacher, become better aware of how you teach as well as how well your students are learning (268-1). By focusing your students learning around a particular task rather than instruction students can gain a better understanding of not just what they’re learning, but why they’re learning it and how it relates to other mathematical ideas.
Introducing a new style of teaching and learning to students could be quite the adjustment. Students who lack experience with opened-ended tasks may be uncomfortable working on a problem they do not immediately know how to solve. A majority of students’ past experiences may have created an inclination to wait until the teacher shows them how to solve such a problem (271-3). This is true for me, and I can expect that in classrooms today as well. Another benefit to evolving your methodology is that by transferring responsibility of understanding from teacher to students, it becomes easier to concentrate on what the students do as opposed to what you are doing the whole time. Freeing up yourself as a teacher will allow you to identify what might associate to the decline of certain tasks.
Two things I question about this article is when they mention reasons for decline, number five states, “Task is inappropriate for a given group of students (e.g., students don not engage in high-level cognitive activities because lack of interest, motivation, or prior knowledge needed to perform…” but I feel like lack of interest and motivation could be a problem in any classroom, regardless of how you teach. I also thought it would be difficult to get a fellow teacher to evaluate me and my classroom management on a regular basis.
The main focus of Mathematical Tasks revolves around the importance of reflection on a teacher’s teaching ability. The research that was done comes from a project called QUASAR which involved teachers in middle schools.(1-2) The focus of the project was to foster mathematical instruction and increase the reflection ability of teachers involved.
It is interesting to see the evolution of the cases involved throughout the articles. Discussed are lesson plans that were not nearly as effective as the teacher believe they could have been.(4-4) A method that comes up is the Mathematical Tasks Framework which describes the process that tasks must go through to allow the students to work on such a task and that is believed to be an important concept in student learning. (3-5) In the case of one of the teachers, the Framework gave her an idea of why her classroom assignment was not as effective as she believed that is should have been.(4-6) It also proved to be useful to judge their classes for how cognitively challenging they were.
Another theme that came up was teachers taking the thinking component out of their assignments by feeding students answers or hints to solve the problem at hand. In the article, there are two attempts at the same lesson. In the first attempt, the teacher failed to cognitively stimulate his students by not fully utilizing his lesson plan and student ability.(5-2) The consequences of the teacher doing something to prevent his students from struggling is indicative of the importance of a well-structured implementation.
After reflecting on the lesson and noticing what he had done wrong in his first lesson, he tried the same lesson plan with another class. By reflecting, the teacher was able to give the proper assistance that did not make his assignment devoid of challenge, was able to get his students to attempt the problem on their own, and had students come up with solutions that he did not expect to see.(5-8) The teacher with the lesson plan went to discussion with another group of teachers, and this time they reflected on why his lesson plan was implemented more effectively. They were able to create a list of factors and conditions that a teacher can utilize to implement classroom tasks with greater proficiency. (7-1)
I believe this article shows the importance of reflection and how important a teacher’s actions or inactions can have on student learning. It goes along with one of the main themes of the previous article Never Say Anything and the fact that the teacher should not deprive their students of learning opportunities. They conclude the article, saying that the students who achieve the best on problem solving are those who were tasked with cognitively demanding tasks which should be the goal of any teacher.
The readings reflections have two main purposes:
1) to hold you accountable for careful reading of and reflection on the readings assigned in class; and
2) to provide you with a record of what you've learned and thought about as a result of the readings.
The readings reflections will be evaluated using the following criteria:
Submit your readings reflection before reading anyone's on the Wiki page and then paste it into the existing reflection page for that current reading. This is due by Wed 9/28 by 11:30pm.Thanks Kaitlin for reminding me and sorry this is late.
No discussion is required but feel free to post if you want to. This is not for extra credit, just if you want to note something.
In the first paragraph of this article the authors made a close connection between reflection on teaching done by teachers and their professional growth (1-1). The people at Quesar suggest that math teachers focus on tasks given in class to reflect on to improve their teaching. They defined a task to be a segment of classroom activity that is devoted to the development of a particular mathematical idea (2-1).
The tasks framework starts with the task as a textbook presents it. Then it moves to tasks as set up by the teacher, and finally the framework deals wit tasks as implemented by students. The article mentions a few instances where teachers’ lessons didn’t go according to the plan. At one point it says that “students appeared overwhelmed” by the number of choices need to be made to begin an assignment (4-3). When students encounter challenges they are many times indecisive in choosing an approach to attack the problem. We need to give students confidence without directly guiding them to the answer. I also think that students need to be more independent in problem solving, but a teacher will need to design lessons that will improve their independence yet remain rewarding if their attempt fails.
Another case in the article talked about a student asking “how do you do this” (5-1) The teacher let it go for awhile and then gave in and gave direction to the class on how to do the problem. This was later repeated but the teacher refrained from giving them advice on what to do. The framework helped him realize his mistake in the first lesson. When he repeated the lesson he was able to improve his teaching. The framework is only a tool of reflection. The article claims that reflection leads to professional growth, which will lead to better teaching. I can agree that reflection is good to improve teaching, but I thought their framework was too rigid. I don’t think that reflection can be put to some formula. I think that it would be difficult to implement this framework into reflection practices in the classroom. I think having peers (specifically teachers with experience) in the same field observe and suggest ideas to you would be a tremendous help to young teachers. I don’t know how feasible this is because schools are giving teachers more classes to teach, which means less time to plan lessons during the school day. I would personally look for other suggestions on reflection before implementing this framework into my classroom.
Mike F
The article, “Mathematical Tasks as a Framework for Reflection: From Research to Practice,” aims to discuss how teachers can reflect on and evaluate their classroom experiences (268-1). I think it is important for teachers to take a step back and reflect on things that are occurring in their classrooms. If we take a good hard look at what is going on we might discover things such as more struggling students than we thought and so on. As promising as they sounds, unfortunately it is not always easy to figure out what we as teachers need to focus on when reflecting and evaluating (268-2). There is good news however, the QUASAR Project has discovered that focusing on mathematical tasks and their phases of classroom use can help teachers during the process of reflecting (268-2).
A task, defined in the article as a period of class activity that is focuses on the progression of a particular mathematical idea (268-2). To me a task seems to be a problem that would be given to a group of students to solve without giving them any specific way of solving the problem. In this respect students can really dive into the problem and discover on their own a way to solve it. This again leads to the idea of cooperative learning which I think is a good way to learning for students.
The article goes on to describe task phases (270-3) in which implementation is the most important stage because it involves the students thinking for themselves and being creative with how to solve problems instead of simply plugging numbers into a formula.
I particularly enjoy the task given to the students in, “The Case of Ms. Bradford” (271-1). It seemed like something the students would really enjoy and get them actively involved but also something they could learn from. However, I also saw that how it could be overwhelming for the students. I think with the right changes it could have been what it was intended. For example, do not give the students as much freedom, I understand it was so the students could come up with ideas on their own but I also believe they need a starting place and apparently so by how overwhelmed they were. I think something of that magnitude needs to be worked up too. Like the article stated, many students are not used to being given that amount of freedom and it slowly needs to be worked up too (271-4).
The discussion of the figure 1 (273-2) I felt that Ron gave his students too much information instead of letting them figure it out. For example he told them to find what percent was 6 shaded squares on a 10X4 rectangle. Instead I think maybe give them a percent and have them figure out the number of squares that would equal. I feel that in working this way it was would been more of a challenge instead of finding the “tried and true” method.
Overall, I think it is important for teachers to reflect on their lessons and find out what is working and what is not. Also once teachers are more comfortable in their classrooms perhaps video tape lessons and discuss them with future teachers to get some advice. I think its hard sometimes to see what is going wrong with a lesson plan that you worked really hard on, which is where another pair of eyes might help.
Kaitlin Froehlke
This article presents teachers with a framework they can use as they reflect on classroom experiences, making them aware of how they teach and how students are learning (1-1). The main focus for the framework is on mathematical tasks (where a task is defined as “a segment of classroom activity that is devoted to the development of a particular mathematical idea” (1-2)).
The day-after-day experience of tasks provides the foundation for what students believe about the nature of mathematics (2-1). Additionally, I believe that as future teachers we have been conditioned by tasks that we did in middle and high school; our own ideas about mathematics come from this cumulative experience. Therefore, we must understand the drastic effects the collective experiences of students in the classroom have in shaping their ideas and perceptions and utilize tasks in our classrooms that cause students to see mathematics for its logic, beauty, and complexity.
There are different levels of demands in certain tasks; lower-level demands include memorization and procedures without connections, whereas procedures with connections and actually doing mathematics fall into the higher-level (3-1). Higher-level demands allow students to attach meaning to their work and apply their knowledge with an answer that makes sense and can be justified (3-2). Clearly, we as teachers should aim to set up tasks that employ the higher level, and this fulfills the goals of NCTM’s standards.
First, as we have seen, teachers should focus on tasks. Next, they should concentrate on task phases, which include the way the tasks appear in instructional materials, the way teachers set them up, and the way students go about working on the task (3-3). These are vital factors in influencing what students actually learn. Sometimes, as the author points out through the case studies in which the framework was applied, tasks (even ones that are set up at higher levels) evolve and do not produce the intended results (3-4). The way students actually carry out the task can change the level of thinking dramatically (3-4). In order to remedy this, teachers apply the framework and use it for reflection.
In keeping with the past several articles we have read, I noticed a few consistent themes: the power of teachers’ questioning and listening skills in shaping students’ learning, our tendency to give answers and help too quickly, and building on prior knowledge (which is reminiscent of part of the learning standard from NCTM). As with the other articles we read recently, this article stressed how this approach is not easy. For the teachers in the case studies, it was easy to show students how to set up the problem and consequently bring a task to a lower level than intended. According to students’ prior knowledge, when they do not understand something, they wait for the teacher to show them how to do it (4-4). “[B]y succumbing to the students’ requests of ‘how to do it’”, we reduce or eliminate “the challenging, sense-making aspects of the ask, thereby robbing students of the opportunity to develop thinking and reasoning skills and meaningful mathematical understandings.” (5-2); I certainly agree.
As we consider mathematical tasks as a way to reflect on our teaching, our reflection becomes more focused and can lead to significant, beneficial change. Using this reflection framework with other teachers yields even better results through cooperation, and their insights can be very helpful as well. As I begin my teaching career and start, even now, to reflect on my teaching, an integral part of this will be to evaluate the mathematical tasks and how the tasks are carried out in my classroom. Without considering the tasks, I may have no idea where to begin to improve my teaching; my students’ learning would suffer from this. Therefore, in order to help my students understand mathematics as well as they possibly can, I must evaluate and reflect upon mathematical tasks in my classroom.
Mandi Mills
Stein and Smith’s article “Mathematical Tasks as a Framework for Reflection,” encourages teachers to reflect upon their own teaching styles in order to improve their abilities as an educator. They point out, however, that simply asking teachers to reflect is too vague, and finding a focal point upon which to base our reflections is the most difficult part of the process (1-2).
This is why Smith and Stein suggest that our focus be on mathematical tasks in the classroom. Students expand their mathematical abilities while working with numerical problem solving tasks (2-2), rather than listening to a lecture or following a set procedure. Because most of their growth as mathematicians happens during their work on mathematical tasks, our ability to design and carry out appropriately challenging tasks is essential. If we reflect upon our teaching abilities in regards to creating these challenging tasks, than we keep our improvement as teachers centered on students’ learning.
The authors go on to suggest that we evaluate our effectiveness in the classroom with QUASAR’s framework best by reviewing previous lessons with colleagues (7-3). I thought this was a clever notion as it mirrors how we group students together in hopes that they will help each other. By working with each other as a community of teachers we will be able to find which of the tasks that we create are already High-Level and which need adjustment to provide students with a better opportunity to think about the problem and reason out an answer that makes sense for them.
In the case of Ms. Bradford, she dealt with a problem that I, myself, have concerns about. How are we to know what tasks are too High-Level? Her tape-roll-toss task lacked direction for the students, who felt overwhelmed by the amount of choices (4-3). This sort of anecdote worries me because while we may plan a lesson that we feel genuinely inspires students to stretch their understanding, if it is not implemented with enough direction than it can fall flat. Though we have discussed at length the benefits of a classroom that minimizes direction, too much open-endedness can leave students feeling frustrated and discouraged by their lack of understanding.
As educators, we must be able to find the balance between too much and too little direction in the classroom. By reflecting on our lessons, with the focus on student learning through mathematical tasks, we can improve the overall atmosphere of our classrooms to one in which students feel capable and interested.
Valerie Gipper
The article “Mathematical Tasks as a Framework for Reflection: From Research to Practice” introduced the idea of tasks used in classrooms. When I was reading this article, I couldn’t help but notice much of what the authors were talking about seemed to be similar to what we have been reading: students need to work things out for themselves.
Examples were shown of two different types of tasks. The first was lower-level tasks. These entailed the students memorizing the use of algorithms and computation in addition to doing multiple of these exercises in one task (3-1). I can remember learning math in this type of way. I recall being given plenty of worksheets which said to find the decimal of this or what percent is this fraction? This is the type of tasks we want to avoid with our students.
The other type of task, the better choice, is high-level. These allow students to go though procedures like they would in lower-level tasks but also building mathematical connections as they work though it (3-2). In the example they showed in figure 1, the students were asked to use pictorials to explore the concepts of percents, decimals, and fractions. This is considered a high-level task where as, shown also in figure 1; the lower level task is simply computation of fractions and such.
The case of Ron Castleman was very interesting and proved to be great insight to the differences between the two tasks. In the first part of this case, Ron attempted to have a lesson similar to that shown in figure 1 in his classroom. The idea was for his students to use a rectangle to find percents, decimals, and fractions. After he started the class he ended up giving his students hints on how to find the answer without using the rectangle. As one of Ron’s colleagues noted, by allowing the students to bypass using the rectangle to find the information, they were completely losing the meaning of a decimal, fraction, and percent (5-3). When I am in a math class and do not understand something my first instinct is to go to the teacher and ask for help. What I gained from this article is the teacher telling a student a huge hint is not helpful. I look back at times when I have asked the teacher for assistance and I realize they either end up doing the entire problem for me or at least give me so many hints I can do it without any more trouble. For the purpose of getting the assignment done this is good for the student but that is where it stops. I don’t remember what the concepts were behind those problems which I asked for. I also have no idea why some of the mathematical techniques work. All I know was I finished the problem. Ron learned to fix this problem by redirecting the students to the original point of the problem, use the rectangle and this helped the students work on their problem solving skills as well as allow them to expand on their connections between mathematical concepts (5-6).
Towards the end of the article, teachers observing one another was discussed. This is a great way to get feedback on how things are going in your classroom (8-2). In addition, the teachers who are being observed will be able to learn from their mistakes and know what seemed to work and what didn’t (8-3). The idea of your co-workers observing and taking notes on what you are doing holds you accountable for actually making sure you are supplying your students with a high level task. In addition, the teacher who is observing you should be able to see where your weaknesses are in the process which is a good to know. In addition, they talked about teachers observing themselves. The suggest videotaping yourself, which allows you to reflect on your own time in an objective manner (8-4). Either form of observation seems to give the same results. The teachers are getting feedback and the students will have a better experience because of it.
Katey Cook
The article, “Mathematical Tasks as a Framework for Reflection” illustrates the point that it is beneficial for teachers or educators to reflect not only on their students’ progress but on their own as well. (1-1) This may seem to be common sense to most but the idea goes a bit deeper than just common sense. Most teachers will have an informal thought of what went well and what needed more work but rarely do they have an in-depth conversation with a colleague for instance to discuss how to improve their teaching style.
So often as we discuss in class mathematics instruction follows the same traditional path where students memorize steps or procedures but do not gain a full understanding of the main concept. We saw evidence of this in class on Tuesday. Fred knew how to perform the operation algebraically but until that light bulb went off he didn’t quite grasp the whole concept. The author found her students were doing the same thing, memorizing operations rather than thinking about the task as its laid out.(2-1)
In the case of Ms. Bradford, what she thought to be a complex lesson plan that would engage her students while having them think at a higher level of problem solving failed during implementation. (4-2) She found that posing an open ended question to her students was a new and scary experience for them which did not help her execute her plan. Although her lesson plan was a good one she adapted quickly so that the students didn’t get dejected and then later reflected on the lesson with colleagues and found other ways to implement her lesson.(4-3) This shows the resolve a teacher needs to have. Realize when a lesson is failing, adapt that lesson so that the students don’t lose interest, and reflect on the lesson and make changes as needed.
Mr. Castlemen’s story is quite similar although I really enjoyed seeing the progression of the lesson plan. At first he laid out an open ended question for the students but when the students quickly began to ask questions he began to fall into the deep dark pit of straight answering those questions.(5-2) After he had felt his lesson failed he went to colleagues and friends to discuss and diagnose the problem with his lesson. The problem became clear but the solution would be the hard part. He decided that this time he would hold off on answering questions and instead ask questions in return to get his students thinking about the problem in another light. (5-6) Although this task took much longer than he had anticipated the outcome of the majority of his students comprehending the material was well worth it.(5-7)
As we become teachers in our own classroom this lesson is one that should be learned early. Not every lesson will work as planned and not every student will understand the concepts we present immediately. It is integral to the students and the educator’s success that the educator constantly reflects on lessons given to improve lessons that still need to be addressed. I think colleagues in our future schools as well as those met during our classes at WMU will be an invaluable resource to an educator who constantly wants to improve. I hope that’s all our goals.
Christopher Cardon
In the article, “Mathematical Tasks”, Stein and Smith discuss the important aspect of reflecting upon lessons and how students are responding. I feel reflection is an essential part of a teacher’s career. If the person’s goal is to be an effective teacher, then being able to critique and learn from your own teaching methods gets you one step closer to that goal. Not only does it benefit the teacher by making her better, but more importantly the students will benefit because they will be experiencing a higher-quality education.
Stein and Smith cover a framework for reflection and the different ways it can be used (2-1). The framework is based on mathematical tasks that are focused on teaching a specific idea. The task can be completed by lower-level or higher-level approaches depending on what the teacher’s goal is for the lesson (3-2). When I was in middle school, I can only remember having low-level demand problems placed in front of me. My teachers back then were more focused on quantity over quality, and it didn’t really matter if we were developing thinking skills in the process (5-3). Now I am conditioned to want to work on lower-level demand problems because I was never introduced to higher-level work. I realize now what an impact these higher-level approaches can have on a student. Not only do they allow students to build connections between mathematical concepts (3-2), but it increases students’ confidence in mathematics. A student who is familiar with higher-level tasks will be more willing to try numerous strategies to obtain a correct answer as opposed to giving up after one try. Whereas, the lower-level task student that is introduced with a higher-level task would probably get frustrated if she couldn’t figure the answer out right away. I am basing that conjecture on articles that we have read and also on my own experience.
Ms. Bradford tried to apply the mathematical tasks framework in her classroom with the tape-roll toss problem (4-2). The students became overwhelmed with the task and what was expected of them, and I actually did too. Just reading about the tape-roll toss made me nervous. If I had been in that classroom, I would have reacted the same way as all those students because of my lack of experience with open-ended tasks. Students that have been trained to solve lower-level tasks become uncomfortable when presented with a problem that they don’t know how to solve (4-4). So I plan on incorporating higher-level tasks into my curriculum. That’s not the way I was taught, but I am realizing there are much more beneficial ways to teach math.
Hailey McDonell
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The article “Mathematical Tasks as a Framework for Reflection: From Research to Practice” stresses the importance of constant self- or colleague-evaluation of teachers. This continual reflection makes us aware of how we are teaching. Is it effective? Is it confusing? They have provided us with a “framework for reflection based on the mathematical tasks used during classroom instruction and the ways in which it has been used by teachers,” (269-1). They have created this because they feel that task form the basis for learning and are varied in level of demand (269-2) The steps in The Mathematical Task Framework include curricular materials, set up by teacher, implemented by students, and when learning occurs (270-4).
The authors go on to give cases to reflect upon based on their research for the QUASAR Project. First, Ms. Bradford challenged her class to create a game board for an upcoming carnival at their school for a ring toss-style game (271-2). She supplied them with various different materials to use and set them free to create. Unfortunately, most of her class was not accustomed to this time of mathematical exploration and appeared overwhelmed by too many options. This example sounds so familiar to many creative projects I had encountered back in high school. For example, in English everyone had to write a research paper, but the topic of the paper could be practically anything school appropriate. Most kids spent the first week complaining that the instructions were too vague and nobody had a clue where to start or what to research. Unlike Ms. Bradford’s choice to guide students in their design (271-3) my teacher stayed neutral and forced us to create something all on our own.
The article then gives the case or Mr. Castleman who felt like his class answered one of his tasks too quickly (271-7). They decided to apply the Mathematics Task Framework to think about the lesson. After observing a videotape of the class, they saw that Mr. Castleman started to guide his class just as Ms. Bradford had. Later, he used the same lesson with a different class and tried to apply some tactics learned from the Framework. This time “students showed no inclination to even check the plausibility of the answers that they came up with against the diagram,” (272-3). This Mathematical Task Framework allows us to use more complex and meaningful tasks that are better for students to make sense of mathematics.
I think videotaping your own teaching to reflect upon what your students are thinking while working on a task is a worthwhile self-assessment tool, and if possible ask a trusted colleague to give you constructive feedback in real time. Real-time observation can benefit you both because it forces your colleague to give critical feedback and not gloss over the important areas that need growth (275-1). Another important tid-bit I got out of this article is the suggestion to ask students to explain their varying paths to the same answer to articulate that there isn’t always ONE right way to answer a mathematical task (272-7).
Tori Ward
The article “Mathematical Tasks” is about the importance of teachers reflecting their work to better their performance in the classroom. According to the professional standards for teaching mathematics, a primary factor in teachers’ professional growth is the extent to which they “reflect on learning and teaching individually and with colleagues”. 268-1 Self reflection is important factor because it helps you as individual acknowledge the things you are not doing so well to help student out in the classroom. Reflection with colleagues is important because they can give helpful strategies for you to use in the classroom. Teachers’ reflecting on their classroom experiences is a way to make teachers aware of how they teach and how their students are thriving within the learning environment that has been provided. Reflection holds the key to improving ones teaching as well as to sustaining lifelong professional development. 268-1 Although self reflection is a good thing a question to consider is figuring out what to focus on that needs improvement.
Therefore in this article there is a framework provided for reflection based on the mathematical tasks used during classroom instruction. A task can involve several related problems that can take up an entire class period. 269-1 The focus on mathematical tasks is built from the idea that the task used in the classroom form the basis for student learning.269-2 All tasks lead to student thinking. The day in and day out cumulative effect of classroom based tasks leads to the development of students’ implicit ideas about the nature of mathematics about whether mathematics is something about which they can personally make sense and about how long and how hard they should have to work to do so. 269-2 There are three phases through which a tasks goes through: first, as they appear in curricular or instructional materials on the printed pages of textbooks, and so on; next as they are set up or announced by the teacher; and finally as they are actually implemented by students in the classroom. 270-5 All these, are especially viewed as important influences on what students actually learn. After teachers learned about the framework, they began to use it as a lens for reflecting on their own instruction and as a shared language for discussing instruction with their colleagues. A teacher named Theresa said that the framework gave her a language for describing events that has occurred in her classroom and for understanding why things may not have worked out as she envisioned that they would when she did a reflection on her classroom. 271-5
When using the framework it should draw attention to what students are actually doing and thinking about during mathematics lessons. The focus on student thinking helps the teacher adjust instruction to be more responsive to, and supportive of, students attempts to reason and make sense of mathematics. 274-2 Factors associated with the maintenance of high level cognitive demands are scaffolding students thinking, that they are monitoring their own progress, sufficient time is allowed for explorations and tasks build on students prior knowledge. More tasks are also listed in the article but these are just a few to name. 274-5 Factors associated with the decline of high level cognitive demands are classroom management problems or the teacher shifts the emphasis from meaning, or understanding to the correctness or completeness of the answer. More tasks are also listed in the article however these are just a few to name. 274-5 Students who performed the best on project based measures of reasoning and problem solving were in classrooms in which tasks were more likely to be set up and implemented at high levels of cognitive demand. These students performed substantial learning gains which is exactly what the NCTM’s professional teaching standards advocated. As a secondary educator I must know how to measure sit and focus on my teaching methods in order to provide a way for my students to obtain a higher level of thinking. With this framework I am able to ask myself what I may have to focus on in order to reach this goal.
Fredrick Martin
Mathematical Tasks as a Framework for Reflection is an article that provided examples of fidelity to the curriculum and your personal planned task. I felt that it is trying to allow us to realize that we must be intentional in planning and thinking about how to help our students without doing it for them in basic terms (6-3). It makes for an extremely helpful rubric when we are trying to evaluate ourselves for the sake of being efficient and maintaining the ability to change to meet the needs of your students. I also realized that there is a certain amount of cooperation amongst the staff that was displayed in the examples of the two teachers whose stories were shared. The three phases (3-4) should probably be examined for consistency in each phase a task goes through. I usually find that if I am inconsistent in my planning or in my delivery it seems that everything I planned on doing goes out of the window and the classroom is free to really do as they please.
This article pushes the need to be intentional and striking the right balance when it comes to doing things in the classroom. Being intentional requires you to think of what kind of cognitive tasks you are requiring of your students and what questions you will ask them (5-5). You don’t want to give them the answers but at the same time you want them to work on the task without giving up on it too easily. Which brings us to how the right balance of different components comes into play. As a teacher we must constantly re-evaluate how much time we need on a task, what kind, if any, extra measures are needed and on and on all day long. I have heard that a teacher makes a decision every 20 seconds. If this is true that doesn’t really leave any time to evaluate themselves. In this case it is nice to have another teacher to be able to critique you on your performance. It is precisely this framework that allows teachers to get the best of both worlds, not only improving their classrooms but also improving themselves.
Denise Slate
This article follows our class’s current theme of how we can incorporate higher-level thinking into our lesson plans. It affirms that direct teaching or lecturing and routine lower level tasks, like memorizing algorithms, are a teaching method of the past. Student centered learning is the focus for teachers today, and the mathematical task framework appears to be a solid method to evaluate your progress transforming your style to a more ideal method.
The framework is basically an organized way to reflect on your classroom experiences in a way that helps you, as a teacher, become better aware of how you teach as well as how well your students are learning (268-1). By focusing your students learning around a particular task rather than instruction students can gain a better understanding of not just what they’re learning, but why they’re learning it and how it relates to other mathematical ideas.
Introducing a new style of teaching and learning to students could be quite the adjustment. Students who lack experience with opened-ended tasks may be uncomfortable working on a problem they do not immediately know how to solve. A majority of students’ past experiences may have created an inclination to wait until the teacher shows them how to solve such a problem (271-3). This is true for me, and I can expect that in classrooms today as well.
Another benefit to evolving your methodology is that by transferring responsibility of understanding from teacher to students, it becomes easier to concentrate on what the students do as opposed to what you are doing the whole time. Freeing up yourself as a teacher will allow you to identify what might associate to the decline of certain tasks.
Two things I question about this article is when they mention reasons for decline, number five states, “Task is inappropriate for a given group of students (e.g., students don not engage in high-level cognitive activities because lack of interest, motivation, or prior knowledge needed to perform…” but I feel like lack of interest and motivation could be a problem in any classroom, regardless of how you teach. I also thought it would be difficult to get a fellow teacher to evaluate me and my classroom management on a regular basis.
The main focus of Mathematical Tasks revolves around the importance of reflection on a teacher’s teaching ability. The research that was done comes from a project called QUASAR which involved teachers in middle schools.(1-2) The focus of the project was to foster mathematical instruction and increase the reflection ability of teachers involved.
It is interesting to see the evolution of the cases involved throughout the articles. Discussed are lesson plans that were not nearly as effective as the teacher believe they could have been.(4-4) A method that comes up is the Mathematical Tasks Framework which describes the process that tasks must go through to allow the students to work on such a task and that is believed to be an important concept in student learning. (3-5) In the case of one of the teachers, the Framework gave her an idea of why her classroom assignment was not as effective as she believed that is should have been.(4-6) It also proved to be useful to judge their classes for how cognitively challenging they were.
Another theme that came up was teachers taking the thinking component out of their assignments by feeding students answers or hints to solve the problem at hand. In the article, there are two attempts at the same lesson. In the first attempt, the teacher failed to cognitively stimulate his students by not fully utilizing his lesson plan and student ability.(5-2) The consequences of the teacher doing something to prevent his students from struggling is indicative of the importance of a well-structured implementation.
After reflecting on the lesson and noticing what he had done wrong in his first lesson, he tried the same lesson plan with another class. By reflecting, the teacher was able to give the proper assistance that did not make his assignment devoid of challenge, was able to get his students to attempt the problem on their own, and had students come up with solutions that he did not expect to see.(5-8) The teacher with the lesson plan went to discussion with another group of teachers, and this time they reflected on why his lesson plan was implemented more effectively. They were able to create a list of factors and conditions that a teacher can utilize to implement classroom tasks with greater proficiency. (7-1)
I believe this article shows the importance of reflection and how important a teacher’s actions or inactions can have on student learning. It goes along with one of the main themes of the previous article Never Say Anything and the fact that the teacher should not deprive their students of learning opportunities. They conclude the article, saying that the students who achieve the best on problem solving are those who were tasked with cognitively demanding tasks which should be the goal of any teacher.
-Marcus Edgette